×

zbMATH — the first resource for mathematics

Stabilization and destabilization of hybrid systems of stochastic differential equations. (English) Zbl 1111.93082
Summary: This paper aims to determine whether or not a stochastic feedback control can stabilize or destabilize a given nonlinear hybrid system. New methods are developed and sufficient conditions on the stability and instability for hybrid stochastic differential equations are provided. These results are then used to examine stochastic stabilization and destabilization.

MSC:
93E15 Stochastic stability in control theory
93D15 Stabilization of systems by feedback
93E03 Stochastic systems in control theory (general)
93C10 Nonlinear systems in control theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[2] Athans, M., Command and control (C2) theory: A challenge to control science, IEEE transactions on automatic control, 32, 286-293, (1987)
[3] Basak, G.K.; Bisi, A.; Ghosh, M.K., Stability of a random diffusion with linear drift, Journal of mathematical analysis and applications, 202, 604-622, (1996) · Zbl 0856.93102
[4] Björk, T., Finite dimensional optimal filters for a class of Itô processes with jumping parameters, Stochastics, 4, 167-183, (1980) · Zbl 0443.60038
[5] Chen, S.; Li, X.; Zhou, X.Y., Stochastic linear quadratic regulators with indefinite control weight costs, SIAM journal on control and optimization, 36, 1685-1702, (1998) · Zbl 0916.93084
[6] Costa, O.L.V.; Assumpcao, E.O.; Boukas, E.K., Constrained quadratic state feedback control of discrete-time Markovian jump linear systems, Automatica, 35, 4, 617-626, (1999) · Zbl 0933.93079
[7] Ghosh, M.K.; Arapostathis, A.; Marcus, S.I., Optimal control of switching diffusions with application to flexible manufacturing systems, SIAM journal on control and optimization, 35, 1183-1204, (1993) · Zbl 0785.93092
[8] Ghosh, M.K.; Arapostathis, A.; Marcus, S.I., Ergodic control of switching diffusions, SIAM journal on control and optimization, 35, 1952-1988, (1997) · Zbl 0891.93081
[9] Grenander, U., Probabilities on algebraic structures, (1963), Wiley New York · Zbl 0139.33401
[10] Il’in, A.M.; Khasminskii, R.Z.; Yin, G., Asymptotic expansions of solutions of integro-differential equations for transition densities of singularly perturbed switching diffusions: rapid switchings, Journal of mathematical analysis and applications, 238, 516-539, (1999) · Zbl 0933.45004
[11] Ji, Y.; Chizeck, H.J., Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE transactions on automatic control, 35, 777-788, (1990) · Zbl 0714.93060
[12] Ji, Y.; Chizeck, H.J.; Feng, X., Stability and control of discrete-time jump linear systems, Control theory and advanced technology, 7, 2, 247-270, (1991)
[13] Khasminskii, R.Z., Stochastic stability of differential equations, (1981), Sijthoff & Noordhoff Alphen & Rijn · Zbl 1259.60058
[14] Khasminskii, R.Z.; Yin, G., Asymptotic behavior of parabolic equations arising from null-recurrent diffusions, Journal of differential equations, 161, 154-173, (2000) · Zbl 0957.35023
[15] Kolmanovskii, V.B.; Myshkis, A., Applied theory of functional differential equations, (1992), Kluwer Academic Publishers Dordrecht
[16] Krishnamurthy, V.; Wang, X.; Yin, G., Spreading code optimization and adaptation in CDMA via discrete stochastic approximation, IEEE transaction information theory, 50, 1927-1949, (2004) · Zbl 1297.94011
[17] Kushner, H.J., Stochastic stability and control, (1967), Academic Press New York, NY · Zbl 0178.20003
[18] Ladde, G.S.; Lakshmikantham, V., Random differential inequalities, (1980), Academic Press New York · Zbl 0466.60002
[19] Liptser, R., A strong law of large numbers for local martingales, Stochastics, 3, 217-228, (1980) · Zbl 0435.60037
[20] Li, X.; Zhou, X.Y.; Ait Rami, M., Indefinite stochastic linear quadratic control with Markovian jumps in infinite time horizon, Journal of global optimization, 27, 149-175, (2003) · Zbl 1031.93155
[21] Mao, X., Stability of stochastic differential equations with respect to semimartingales, (1991), Longman Scientific and Technical New York · Zbl 0724.60059
[22] Mao, X., Exponential stability of stochastic differential equations, (1994), Marcel Dekker New York · Zbl 0851.93074
[23] Stochastic differential equations and their applications, (1997), Horwood Publisher Chichester
[24] Mao, X., Stability of stochastic differential equations with Markovian switching, Stochastic process and their applications, 79, 45-67, (1999) · Zbl 0962.60043
[25] Mao, X.; Matasov, A.; Piunovskiy, A.B., Stochastic differential delay equations with Markovian switching, Bernoulli, 6, 1, 73-90, (2000) · Zbl 0956.60060
[26] Mariton, M., Jump linear systems in automatic control, (1990), Marcel Dekker New York
[27] Mohammed, S.-E.A., Stochastic functional differential equations, (1986), Longman Scientific and Technical New York · Zbl 0584.60066
[28] Sethi, S.P.; Zhang, Q., Hierarchical decision making in stochastic manufacturing systems, (1994), Birkhäuser Boston · Zbl 0923.90002
[29] Shaikhet, L., Stability of stochastic hereditary systems with Markov switching, Theory of stochastic processes, 2, 18, 180-184, (1996) · Zbl 0939.60049
[30] Skorohod, A.V., Asymptotic methods in the theory of stochastic differential equations, (1989), American Mathematical Society Providence
[31] Sworder, D.D.; Rogers, R.O., An LQ-solution to a control problem associated with solar thermal central receiver, IEEE transactions on automatic control, 28, 971-978, (1983)
[32] Yin, G.; Zhang, Q., Continuous-time Markov chains and applications: A singular perturbation approach, (1998), Springer New York · Zbl 0896.60039
[33] Yin, G.; Liu, R.H.; Zhang, Q., Recursive algorithms for stock liquidation: A stochastic optimization approach, SIAM journal on optimization, 13, 240-263, (2002) · Zbl 1021.91022
[34] Willsky, A. S., & Levy, B. C. (1979). Stochastic stability research for complex power systems. DOE Contract, LIDS, MIT, Report ET-76-C-01-2295.
[35] Zhang, Q., Stock trading: an optimal selling rule, SIAM journal on control and optimization, 40, 64-87, (2001) · Zbl 0990.91014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.